nLab higher monadic descent

Contents

Context

Locality and descent

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher algebra

Contents

Idea

Higher monadic descent is the generalization of the notion of monadic descent from category theory to higher category theory. It relates to the descent of ∞-stacks as ordinary monad descent relates to stacks.

See also cohomological descent.

Examples

Amitsur complex, Sweedler corings, Hopf algebroids

For ϕ:BA\phi \,\colon\, B \longrightarrow A a homomorphism of suitable monoids, there is the corresponding pull-push adjunction (extension of scalars \dashv restriction of scalars) on categories of modules

(() BAϕ *):Mod Aϕ *() BAMod B. \big( (- )\otimes_B A \dashv \phi^\ast \big) \;\colon\; Mod_A \underoverset {\underset{\;\; \phi^\ast \;\;}{\longrightarrow}} {\overset{(-)\otimes_B A}{\longleftarrow}} {\bot} Mod_B \,.

The bar construction of the corresponding monad is the corresponding Amitsur complex.

(e.g. Hess 10, section 6)

References

2-categorical monadic descent

A pseudomonadicity theorem for pseudomonads is proved in Theorem 3.6 of:

and sharpened in Theorem A.2 of:

(see Remark A.3 for a comparison with the work of Creurer et al.).

More generally, this paper discusses a notion of 2-fibered categories, originally defined in Gray’s work, and then again introduced and discussed at length in this paper. The appendix also discusses a 2-categorical version of Beck-Chevalley condition needed for a comparison with 2-monadic descent.

(,1)(\infty,1)-categorical monadic descent

A comprehensive treatment in the context of (∞,1)-category-theory, general theory of (∞,1)-monads and their monadicity theorem is in

later absorbed as

Unfortunately, Kontsevich‘s monadicity theorem (July 2004) which is in the setup of A-∞-categories, still remains unpublished. The triangulated version is in Rosenberg’s lectures

  • A. L. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57 pdf, page 36-37.

Its proof is based on Verdier's abelianization functor.

See also, for another point of view,

Discussion of monadic descent for simplicially enriched categories is in

and for quasi-categories in

Last revised on April 5, 2023 at 20:57:15. See the history of this page for a list of all contributions to it.